Gains¶

The gain sets how much weight the learners put on the newest observation when they update their beliefs. With a gain \(g_t\) and regressors \(z_{t-1}\), recursive least squares (RLS) is

\[ R_t = R_{t-1} + g_t\left(z_{t-1} z_{t-1}' - R_{t-1}\right), \qquad \phi_t = \phi_{t-1} + g_t R_t^{-1} z_{t-1}\left(v_t - \phi_{t-1}' z_{t-1}\right), \]

equation by equation, each with its own \(R\). A constant gain \(g\) means the learners weight past data with a geometric decay; \(1/g\) is roughly their memory in quarters (0.05: about five years).

Option	Values (default first)	Meaning
`gain.rule`	`'constant'`, `'koz'`, `'decreasing'`	Constant gain, the endogenous gain of Kostyshyna, Özden and Zhang (2024; KOZ below), or a decreasing gain. The name is a function in `toolbox/+hetlearn/+gain/`. Project baseline: `'koz'`
`gain.value`	`0.0585`	The constant gain (Özden 2025, posterior mean), and the start of `'koz'`. In (0, 1]. Project baseline: 0.05
`gain.by_var.<x>`	number	Gain of one variable: with `'constant'` its gain, with `'koz'` and `'per_variable'` its start value (ignored with a warning under `'common'`)
`gain.koz.mu`	`0.001`	Step size of the gain process; 0 gives the constant gain exactly
`gain.koz.gamma0`	`[]`	Start gain; `[]` = `gain.value`
`gain.koz.Rg0`	`0`	Start value of \(R^g\)
`gain.koz.bounds`	`[0.001 0.25]`	The gain stays inside; a `gain_bound` event each quarter a bound binds
`gain.koz.error`	`'rls'`, `'forecast'`	Signed error that drives the gain: the residual of the perceived law of motion, one step ahead, or the learners' expectation error \(x_t - f^{BR}_{t-1}\) (BR, for bounded rationality, labels the learners)
`gain.koz.error_scale`	`1`	Factor on that error; 1 = model units (percent per quarter for the Smets and Wouters (2007) model)
`gain.koz.structure`	`'common'`, `'per_variable'`	One gain process for every equation of the perceived laws of motion, or one per equation driven by its own error
`gain.koz.master`	`''`	`'common'` only: the variable whose error drives the gain; `''` = the first learning variable; a list = the mean of their errors, so a quarter in which any of them is missing skips the gain update
`gain.decreasing.t0`	`0`	`'decreasing'` only: the initial beliefs count as `t0 + 1` observations; >= 0

The endogenous gain¶

KOZ (2024, eq. 9) use the adaptive step-size algorithm of Kostyshyna (2012). After \(x_t\) is observed, with \(e_t\) the signed error (times error_scale) and \(P\) the projection on bounds:

\[ \gamma_t = P\left[\gamma_{t-1} + \mu R^g_{t-1} e_t\right], \qquad R^g_t = (1 - \gamma_{t-1}) R^g_{t-1} + e_t . \]

The RLS update at the end of t uses \(g_t = \gamma_{t-1}\). The gain rises when errors keep the same sign and falls when they alternate. It updates from the observed error whatever the projection facility does with the beliefs.

Things to know:

mu = 0 reproduces the constant gain bit for bit.
mu is scale dependent: the error enters both equations, so mu acts roughly with the square of error_scale. An error_scale of 4 (annualized percent) works like a 16 times larger mu.
With 'koz', res.learning.<x> also holds gain_R (\(R^g_t\)).
KOZ's Table 1 "R0 = 0.001" is the initial RLS moment matrix, not \(R^g_0\). It is set with beliefs.init.method = 'user' and only beliefs.init.values.<x>.R (example below).

The decreasing gain¶

With gain.rule = 'decreasing' the gain falls with the number of updates, as in recursive least squares with equal weights on all data (Evans and Honkapohja 2001, ch. 2). In the k-th quarter of learning (k = 1 at beliefs.init.date)

\[ g_t = rac{1}{t_0 + k + 1}, \]

with \(t_0\) = gain.decreasing.t0. The initial beliefs count as \(t_0 + 1\) observations, so with the default \(t_0 = 0\) the gain is \(1/t\), with \(t\) the number of observations including the initial beliefs: 1/2 at the first update, 1/3 at the second. The first update thus averages the initial moment matrix \(R_0\) and the first data point, and \(R_t\) stays invertible for a PLM with several coefficients. With a constant-only PLM the belief is the running mean of the data and the initial belief.

Things to know:

The gain goes to zero. Learning from 1966Q1, the gain is 1/227 at the 2022Q2 origin, so the beliefs barely move over the forecast.
A quarter with missing data for an equation (event rls_missing) does not count.
The count carries on from the history into the forecast, density draws and evaluation origins, whatever the projection facility does with the beliefs.
gain.value and gain.by_var are not used.

Examples¶

% Constant gain at the toolbox default
cfg = a1_learn(); cfg.id = 'a1_const0585';
cfg.gain.rule = 'constant';
cfg.gain.value = 0.0585;

% Constant gain 0.05, the baseline's start gain (model variant v4_const)
cfg = a1_learn(); cfg.id = 'v4_const';
cfg.gain.rule = 'constant';
cfg.gain.value = 0.05;

% A different constant gain for one variable
cfg = a1_learn(); cfg.id = 'a1_pinf_gain';
cfg.gain.rule = 'constant';
cfg.gain.by_var.pinf = 0.02;

% Endogenous gain with a larger step and the learners' forecast error as the driver
cfg = a1_learn(); cfg.id = 'a1_koz_fast';
cfg.gain.rule = 'koz';
cfg.gain.value = 0.05;
cfg.gain.koz.mu = 0.01;
cfg.gain.koz.error = 'forecast';

% One gain process per equation
cfg = a1_learn(); cfg.id = 'a1_koz_pervar';
cfg.gain.koz.structure = 'per_variable';

% One common process driven by inflation and wage errors
cfg = a1_learn(); cfg.id = 'a1_koz_master';
cfg.expectations.variables = {'pinf', 'w'};
cfg.gain.koz.master = {'pinf', 'w'};

% Decreasing gain 1/t, counting the initial beliefs as one observation (model variant v4_decreasing)
cfg = a1_learn(); cfg.id = 'v4_decreasing';
cfg.gain.rule = 'decreasing';

% Decreasing gain with initial beliefs worth 20 quarters: first gain 1/21
cfg = a1_learn(); cfg.id = 'a1_decreasing_t0';
cfg.gain.rule = 'decreasing';
cfg.gain.decreasing.t0 = 19;

% KOZ's initial moment matrix R0 = 0.001*I (AR(1): two coefficients)
cfg = a1_learn(); cfg.id = 'a1_koz_R0';
cfg.beliefs.init.method = 'user';
cfg.beliefs.init.values.pinf.R = 0.001*eye(2);

The gain after a run:

res = hetlearn.run(a1_learn(), 'save', false);
g = res.learning.pinf.gain;
g(qq(2022, 2))                       % 0.0499
[min(g(qq(1966,1):qq(2022,2))), max(g(qq(1966,1):qq(2022,2)))]

In the project¶

The baseline KOZ gain starts at 0.05 in 1966Q1 and stays within 0.047-0.052 over 1966Q1-2026Q2 (mean 0.049); it never hits a bound. It is 0.0499 at the 2022Q2 origin and averages 0.0489 over the forecast. Model variant 4 holds it at 0.05:

	`a1_learn` (KOZ)	`v4_const` (0.05)
Average inflation 2022Q3-2026Q2 (%)	3.45	3.50
Learners' share at the end of 2022Q2	0.591	0.595
Largest difference in \(\beta\), 1966-2026		0.012
Largest difference in the share, 1966-2026		0.004

With mu = 0.001 and errors in percent per quarter the gain moves little, so the endogenous and the constant gain give nearly the same results in the Smets and Wouters model.

The decreasing gain (v4_decreasing) differs more. Its learners weight every quarter since 1966Q1 equally, so at the 2022Q2 origin their gain is 1/227 and their AR(1) coefficient is 0.91, against 1.06 under the endogenous gain. Average inflation over 2022Q3-2026Q2 is 3.12 percent and the learners' share at the end of 2022Q2 is 0.618. Over the forecast the coefficient stays between 0.899 and 0.905.